WALTER LITTMAN BO LIU kind of problem is related to the deformation of a membrane with strings attached to several sides of the polygon. In other word

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1 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POLYGONAL DOMAINS Walter Littman Bo Liu School of Math., University of Minnesota, Minneapolis, MN55455 Abstract. The regularity and singularity of variational solutions of problems u u = f in ; T u = u g on ; u = 0 on ; = 0 on 3 with suitable compatibility conditions at vertices of for bounded polygonal domains R are studied by combining Grisvard's (cf. Grisvard[4]) results with perturbation theory and the method of continuity. The variational solutions are proved to be in H () H ( ) for certain geometric polygonal domains. The singular decomposition forms of the variational solutions are given implicitly and explicitly for other geometric polygonal domains.. Motivation. We here consider the following problem 8 >< >: u u = f in (:a) T u = g on (:b) u =0 on (:c) u =0 on 3 (:d) where R is a bounded open polygonal domain, ; ; 3 are a part of and each of them consists of several sides of the polygon, mes( ) 6= 0, T >0is a possibly dierent constant on dierent sides of, is the external normal direction of boundary and is the tangential direction of boundary taken in counterclockwise direction. This kind of problem arises in the study of boundary control theory (see Littman[7][8]) where we need the compactness properties of the solution operator of equations (.a)-(.d). Physically this Typeset by AMS-TEX

2 WALTER LITTMAN BO LIU kind of problem is related to the deformation of a membrane with strings attached to several sides of the polygon. In other words, we need to know the regularity and singularity properties of solutions u of equations (.a)-(.d) near those corners. In [4], Grisvard used Fredholm Properties systematically to study the behavior of solutions of Laplace's equation with Dirichlet, Neumann or mixed boundary conditions near the corners of a polygonal domain. It is well-known that the variational solutions of Laplace's equation with Dirichlet, Neumann or mixed boundary conditions for f L () belong to H () when the polygonal domain satises certain geometrical conditions. Otherwise the variational solution may have singularities near those corners independently of the smoothness of f. The decomposition for the singular part is described in Grisvard[4]. The same results for general elliptic equations were also obtained by Banasiak & Roach[]. In [9] Mghazli investigated the regularity and singularities of solutions of the Laplace equation with Dirichlet-Robin boundary conditions where one actually has u H = () whenever variational solutions exist. The problems (.a)-(.d) are quite dierent from the problem mentioned above because of the term T u on where trace imbedding theorems could not guarantee uj H ( ) when u H (). It is easy to see that if the variational solutions of (.a)-(.d) with proper compatibility conditions at vertices are in H (), then we must have uj H ( ). However u H () turns to be very dicult to establish. Can we expect that vartional solutions u belong to H () and uj H ( ) when (f; g) L () L ( )? The answer is yes under certain conditions. Can we exclude singularities near the corners adjacent to due to the stronger eect of the term T u acting on? The answer is negative as we shall see that the variational solutions could possibly have singularities at each corner of.

3 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL The outline of this paper is as follows. In xweinvestigate the existence and interior regularity of the variational solutions of (.a)-(.d) with compatibility conditions at the corners. A density property needed for reaching our goal will be studied in x3. In x4 a priori estimates for solutions of (.a)-(.d) with compatibility conditions at the corners are obtained by using the Nirenberg- Gagliardo interpolation inequality. Finally in x5, the results of regularity are rst proved by perturbation theory, the method of continuity and Grisvard's results. Then an extension theorem for (.a)-(.d) is derived and a decomposition for the singular part of the variational solutions is obtained.. Existence. We rst introduce some notation to be used throughout this paper. Let m be the number of corners of. Let A j ; j = ; :::; m, denote the vertices of the domain numbered in counterclockwise order and e j ; j = ; :::; m, denote sides of such that e j = (A j ;A j ). We assume A m+j A j and e m+j e j modulo m. Let! j ; j = ; :::; m, denote interior angle of A j. Let j ; j =; :::; m, denotes the tangential direction (counterclockwise) along the side e j. Let j ; j = ; :::; m; denotes the external normal directions on e j. We divide the set f; :::; mg into three subsets D; N and S such that D = fj; e j g, N = fj; e j 3 g and S = fj; e j g. We consider the existence of the variational solutions of equations (.a)-(.d) rst. We need to assume the following compatibility conditions at the corners A j for a smooth function u. I): u(a j )=0; 8A j \ : (:) II): If (j; j +) SN, then either u(a j )=0or u (A j ) = 0 but only u (A j )=0if! j = or 3. If (j; j +)NS,then either u(a j)=0or u + (A j )=0but only u + (A j )=0if! j = or 3. (:)

4 4 WALTER LITTMAN BO LIU III): Either T j u (A j ) T j+ u + (A j )=0 or u(a j ) = 0 for (j; j +) S S where T j for j S denotes the constant T on e j. (:3) We also introduce some notation to be used throughout the paper. L ( )=fv; vj ej L (e j ) 8jSg: H ( )=fv; vj ej H (e j ) 8jSg: V =f(v; p) H () H ( ); v =0 on ; p = vj ;p(a j )=0 if the compatibility conditions require v(a j )=0 in (:) (:3)g: The associated inner product on L ( ); H ( ) is naturally inherited from the inner products on L (e j ), H (e j ) respectively. The associated inner product on the test Hilbert space V is then inherited from the inner products on H () and H ( ). We say v V if (v; vj ) V. Weconsider the following variational problem: Find u V such that OuOvdxdy + T u ds = fvdxdy + gvds 8v V : (.4) To prove the existence and \uniqueness" of the variational solutions u of (.4), one has to show that the left side of (.4) denes an equivalent inner product on V or on V =P 0 which denotes the quotient space of V modulo constants. We need Lemma.. Let F 0 ::::F k0 be the bounded linear functionals on the space W k;p (), p. If q P k where P k is the polynomial space of degree k of two variables such that we have q =0whenever F (q) =F (q)= = F k0 (q)=0, then kvk W k;p () C[jvj W k;p () + k X 0 i=0 jf i (v)j] (.5)

5 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL where jvj W k;p () is the seminorm of the space W k;p (). Proof. Suppose that (.5) is not true. Then there is a sequence v n W k;p () such that kv n k W k;p () = 8n; but lim n! [jv nj W k;p () + k X 0 i=0 jf i (v n )j] =0: (.6) Therefore there is a subsequence, whichwe still denote by v n, such that lim n! v n = ^v weakly in W k;p () since W k;p () is weakly compact. One has lim n! v n = ^v in W k ;p () by Rellich's theorem (cf Adams[]). From (.6) one easily has P j^vj W k;p () + k 0 i=0 jf i (^v)j = 0. Therefore ^v = 0 by assumptions. Then it follows that v n converges to 0 in W k;p () strongly since lim n! jv nj W k;p () =0. Contradiction to (.6). Q.E.D. Theorem.. For any (f; g) L () L ( ), there is a variational solution u V such that (.4) holds and kuk V C[kfk L () + kgk L ( ) + kuk L ( )]: (.7) Moreover if mes( ) 6= 0 or if either uj ej (A j ) = 0 or uj ej (A j j S (uj ej (A j ) = kuk L ( lim Ae j ;A!A j uj ej (A), same for uj ej (A j ) on the right side of (.7) can be dropped. )=0for some )), then the term Proof. Consider the functional F (v) = R e j vds for some j S. From lemma. we have kvk H () C[kOvk L () + jf (v)j]. It follows from RR Poincare's inequality and the trace imbedding theorem (cf. Adams[]) that jouj dxdy + R T j u d Ckuk V if mes( ) 6= 0 or if u(a j ) = 0 for some j S and other- RR wise R jouj dxdy + (cf. Gilbarg & Trudinger[3]), we conclude that theorem. holds. T j u ds Ckuk V =P 0. From the Lax-Milgram theorem Q.E.D.

6 6 WALTER LITTMAN BO LIU Remark.: From the proof of theorem. we also know that all of the variational solutions of (.4) are the same up to a constant and would have the same regularity and singularity properties. The following theorem tells us about the local regularity of the variational solutions of (.4). Theorem.. Let (f; g) L () L ( ) and u be a variational solution of (.4). Then u H loc ( [ e k) H loc (e j); 8j S and all k. Furthermore u satises (.a)-(.d). Proof. It is well known that (.a), (.c) and (.d) hold as well as u H loc ([ e j ) 8j D [ N. It is sucient to check the regularity of u near e j ; 8j S. Without loss of generality, we assume that e j where j S is xed and lies on the x-axis. Let x 0 e j. Consider the upper semiball B + R (x 0) ( [ e j ) with center x 0 and radius R. For any R 0 < R, let e 0 j = B R 0(x 0) \ e j and M h (u) = u(x+h;0) u(x;0) where 0 <h<r R 0. Then by (.4) one has h j B + R0 (x 0) = j B + R0 (x 0) [M h (Ou)] Ovdxdy + e 0 j Ou[M h (Ov)]dxdy + T M h ( u x ) x dxj e 0 j T u x Mh ( x )dxj C[kfk L () + kgk L ( )][kvk H (B + R0 (x 0)) + kvk H (e 0 j ) ] 8v C 0 (B R 0(x 0 )): Thus k M h (Ou)k L (B + R0 (x 0)) + k Mh u x k L (e 0 j ) C[kfk L () + kgk L ( )]: (.8) From (.8) and lemma 7.4 in Gilbarg & Trudinger[3], one has u x, L (B + R 0 (x 0 )) and uj ej H (e 0 j ). Thus u y Q.E.D. L (B + R 0 (x 0 )) by (.a). u xy

7 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL The above theorem tells us that we just need to look at the regularity of the variational solutions of (.a)-(.d) with compatibility conditions (.)- (.3). For those corners which are not related to (i.e. those corners not next to the string part), one can use polar coordinates near the corners and the method of separation of variables to investigate regularity(cf. Grisvard[4]). This technique would not work for the corners related to because the operator, (0;! j ) for some j or j +S,with boundary conditions at =0;! j is no longer self-adjoint. 3. Density Property. In this section we focus on a density property of the Hilbert space V = f(v; p) H () H ( ); v =0 on ; p = vj ; =0 on 3; v satises (.)-(.3)g with the associated norm k k V = kk + H () kk H (. We sometimes ) nd it convenient by slight abuse of notation to say v V if (v; vj ) V. Grisvard[4] proved that H k () \V is dense in V when [ 3 =. However that H k () \V is dense in V in general is not obvious. If one proves that H 3 () \V is dense in V, then H k () \V is obviously dense in V. Here we have Theorem 3.. For any k 3, H k () \V is dense in V. Proof. Let B = f k g j k k=0;;jm be the trace operators and B(V ) be the image of the trace operator B. From Grisvard's argument (cf. Grisvard[4] theorem.6.) and the Hahn-Banach theorem, it is sucient to show that B(H k ()\V ) is dense in B(V ). Obviously B(V ) is a subspace of js[h (e j ) H = (e j )] jd[n[h 3= (e j ) H = (e j )]. Let m j= (g j;h j ) B(V ). Then m j= (g j;h j ) satises (g j ;h j )H (e j )H = (e j )ifjsand (g j ;h j ) H 3= (e j )H = (e j )

8 8 WALTER LITTMAN BO LIU if j D [ N and g j =0 on e j if j D (3.) h j =0 on e j if j N (3.) g j (A j )=g j+ (A j ) j =;; :::; m (3.3) gj(a 0 j ) [ cos! j g 0 j+ + sin! j h j+ ](A j ) j =;; :::; m (3.4) h j (A j ) [sin! j g 0 j+ + cos! j h j+ ](A j ) j =;; :::; m: (3.5) The symbol 0 denotes the tangential derivative along e j taken counterclockwise and the symbol means that when at most two of fgj(a 0 j ); h j (A j ), g 0 j+ (A j); h j+ (A j )g for the A j are not well dened, we can redene their values at A j such that (3.4) and (3.5) for the xed j become identities. Checking B(H k () \ V ) carefully, one has that B(H k () \ V ) is a subspace of j H k = (e j ) H k 3= (e j ) and m j= (g j;h j ) satises the following equation as well as (3.)-(3.5). [ cos! j ^g 00 j sin! j^h0 j](a j ) [ cos! j ^g 00 j+ + sin! j^h0 j+](a j ): (3.6) Therefore we just need to check if there are smooth functions (^g j ; ^h j ) and (^g j+ ; ^h j+ ) satisfying (3.)-(3.6) locally to approximate (g j ;h j ) and (g j+ ;h j+ ) in corresponding trace spaces for each corner A j. The cases (j; j +) DD[DN[ND[NN were done by Grisvard [4] (Theorem.6.). We here discuss the case j S. Case : j +D. Then g 0 j+ = 0 on e j+. If! j =, one has g j (A j ) = g 0 j (A j) = 0 and h j (A j )=h j+ (A j )by (3.)-(3.5). If! j 6=, one has g j (A j )=0, h j (A j )= cot! j g 0 j (A j) and h j+ (A j ) = csc! j g 0 j (A j)by (3.)-(3.5) and (.). Thus one can choose smooth functions ^g j, ^h j, ^h j+ with the above compatibility conditions at A j and ^g (i ) j (A j )=^g (i ) j+ (A j)=^h (i ) j (A j )=^h (i ) j+ (A j)=0 (3.7)

9 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL where i, i to approximate g j, h j, h j+ in H (e j ), H = (e j ), H = (e j+ ) respectively. Case : j + N. Then h j+ =0. If! j = = or3=, one has g 0 (A j j)=0 and h j (A j ) = sin! j g 0 j+ (A j). If! j 6= = or3=, one has g j+ (A j ) = sec! j g 0 (A j j), h j (A j ) = tan! j g 0 (A j j) by (3.4)-(3.5). g j and g j+ also satisfy the compatibility condition (.) and (3.3). Therefore one can choose smooth functions ^g j, ^h j, ^g j+ which satisfy (3.7) and the above compatibility condition at A j to approximate g j, h j, g j+ in H (e j ), H = (e j ), H 3= (e j+ ) respectively. Case 3: j + S. If! j =, then g 0 (A j j) = g 0 j+ (A j) (= 0 if the rst equality isrequired and T j 6= T j+ in (.3)) and h j (A j )=h j+ (A j ). Otherwise one has h j (A j ) = cot! j g 0 (A j j) csc! j g 0 j+ (A j), h j+ (A j ) = csc! j g 0 (A j j)+ cot! j gj+(a 0 j ) by (3.4)-(3.5). g j and g j+ also satisfy the compatibility condition (.3) and (3.3). Once again, one can have smooth approximations of g j, h j, g j+, h j+ with the above compatibility conditions at A j and (3.7) in H (e j ), H = (e j ), H (e j+ ), H = (e j+ ) respectively. Q.E.D. 4. A Priori Estimate. Lemma 4.. Assume that is a bounded open polygonal domain, 0 and t (0; ]. If v V,then there is C() independent of and t such that ( v xy ) dxdy + v v x y dxdy + T ( v ) ds t T ( T t v + t v) ds + C() ( ) ds: (4.) Proof. By theorem 3. it is sucient to prove that lemma 4. holds for v H 4 () \V. Using integration by parts with respect to y rst and then x, one

10 0 WALTER LITTMAN BO LIU has ( v xy ) dxdy = v v x y dxdy x ( y )ds: (4.) Using integration by parts with respect to x rst and then y, one has (4.) and (4.3) give ( v xy ) dxdy = ( v xy ) dxdy = v v x y dxdy + y ( x )ds: (4.3) v v x y dxdy + [ y ( x ) One has the following relationships of coordinate substitution. 8 >< >: x y = = where =( ; ). Using (4.5), one has = [ y ( x ) mx j= e j [ v X m x ( y )]ds = j= Consider the term m P v ]ds = j j= j A j A j =0. Let j S. Case : j +D. If! j one has (A j )= + [ ( e j y x ) mx j= e j v j ds + x ( y )]ds: ( x y )]ds mx j= j A j A j in (4.6). If j D[N, then R e j v j =, then (A j ) = cot! j (A j ) by (3.4) and (3.5). (4.4) (4.5) j A j A j : (4:6) ds = + (A j ) = 0. Otherwise Case : j + N. If! j = or 3, then (A j ) = + (A j ) = 0. Otherwise one has (A j ) = tan! j (A j ) by (3.4) and (3.5).

11 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL Case 3: j +S. If! j =, then (A j )=0. Otherwise one has + + (A j ) + (A j ) = cot! j+ + (A j ) + csc! j (A j ) and (A j ) = [cot! j + csc! j + ](A j ). No matter which case it is, one has the following inequality if one applies a special case of the Nirenberg-Gagliardo interpolation inequality (cf. Adams[], Lemma 5.8 or Henry[5]) to v 0 (s) one j and Young's inequality j mx j= j A j A j X jc() [j (A j )j + j (A j )j ] js C() X js C(;)k k L ( k k H (e j ) k k L j (e j ) j + ) ( v ) ds (4:7) where >0can be arbitrary small. We also have j T vdsj t T ( ) ds + v t vds = t ( ) ds 0: tt v ds; (4.8) If we choose suciently small such that R ( v ) ds R then (4.4) and (4.6)-(4.8) provide (4.). Q.E.D. T ( v ) ds, The term k k L () on the right side of (4.) can not be dropped in general except if 0 > 0 or if has special geometry. We now have Theorem 4.. Assume that is a polygonal domain, 0 and t (0; ]. If v V,then there is constant C() independent of t such that kvk H () + kvk H ( ) C()[kvk L () + k T t v + t vk L ( ) + kvk L ( )]: (4.9)

12 WALTER LITTMAN BO LIU Moreover if mes( ) 6= 0or if either u(a j )=0or u(a j )=0for some j S or if > 0, then the term kvk L ( ) can be dropped from the right side of (4.9). Proof. From lemma 4. we have jvj H () + jvj H ( ) C[k v x k L () + k v y k L () +k v xy k L () + C()[kvk L () + k T t v + t vk L ( T ( v ds] ) + k ) k L ( )]: (4:0) The term k k L ( ) in (4.0) can be absorbed in kvk H () + kvk H ( ) which is equivalent to kvk V. Thus (4.9) follows from (4.0) and inequality similar to (.7). Q.E.D. Remark 4.: Theorem 4. holds even if T is a piecewise smooth function on such that 0 <T 0 T T <+on where T 0 and T are constants. Corollary 4.. Assume that is a polygonal domain and that u V is a variational solution of (.4). Then kuk H () + kuk H ( ) C()[kfk L () + kgk L ( ) + kuk L ( )]: (4.) Moreover if mes( ) 6= 0or if either u(a j )=0or u(a j )=0for some j S, then the term kuk L ( ) can be dropped from the right side of (4.). 5. The Continuity Method and Singularity Analysis. In this section we always assume that is a bounded open polygonal domain. We rst quote the method of continuity. Lemma 5. (cf. Theorem 5. of Gilbarg and Trudinger[3]). Let B and B be Banach spaces. L t : B!B is bounded linear operator for every t [a; b].

13 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL If there is a constant C independent of t such that kvk B CkL t vk B 8t [a; b]; 8v B (5.) and L t is uniformly continuous in L(B ; B ) with respect to parameter t on [a; b], then L b maps B onto B if and only if L a maps B onto B. As an application of lemma 5., we consider L t : V! L () L ( ) dened by L t (v) =( 4v; t T v +v); 8t [0; ] for some > 0. From theorem 4. and remark. we know that L maps V onto L ()L ( ) if and only if L t0 maps V onto L () L ( ) for some small parameter t 0. However using perturbation theory, we claim Theorem 5.. Let > 0. If the following problem 8 >< >: u = f in T u + u = g on u =0 on ; u =0 on 3 with compatibility conditions (.)-(.3) has unique solution u V (5.) for any (f; g) L () L ( ), then the variational solutions of (.a)-(.d) with (.)-(.3) belong to V for any (f; g) L () L ( ). Proof. From the assumption we know that L 0 maps V - and onto L () L ( ). So L 0 maps L () L ( ) onto V. Moreover kl 0 (f; g)k V C()[kfk L ()+kgk L ( )]. Thus kl 0 (L 0 L t )k L(V ;V ) tkl 0 (0; )k L(V ;V ) t C(). Therefore we know that there exists a t 0 > 0 such that L t 0 = [I L 0 (L 0 L t0 )] L 0 exists. Q.E.D. It is easy to see that the regularity of the solution of (5.) can be reduced to the regularity of the solution of (5.) with homogeneous Dirichlet boundary condition on instead of the second equation of (5.). Using polar coordinates,

14 4 WALTER LITTMAN BO LIU the Fredholm alternative theorem and the fact that with Dirichlet, Neumann or mixed boundary conditions is a self-adjoint operator and has compact resolvent, Grisvard gave a complete study on the regularity and singularities of the solution of (5.). Using theorem 5. and the results of theorem.3.7 in Grisvard[4], we have the following regularity result. Corollary 5.. If 0 <! j when (j; j +) (D[S)(D[S) or N N and 0 <! j when (j; j +)(D[S)N or N (D [ S) for any j =; ::::m, then variational solutions of (.a)-(.d) with compatibility conditions (.) and (.) belong to V. Can that the variational solutions of equations (.a)-(.d) with compatibility conditions (.)-(.3) belong to V but the solution of equations (5.) with compatibility conditions (.)-(.3) not belong to V? We found from the proof of theorem 5. that it is possible that kuk V is uniformly bounded by kl t k L ()L ( ) for small parameter t and thus the proof of theorem 5. is reversible. For the rest of this section we assume that contains no two consecutive sides e j and e j+ for some j. We rst have an extension theorem. Theorem 5.. If > 0 is a constant, then there is t 0 > 0 (depended on the geometry of ) such that for any 0 < t < t 0 and g L ( ), there exists a G V (see notation at the beginning of x3)such that t G. Moreover T G + G = g on kgk V C()kgk L ( ) (5.3) where C() is independent of parameter t.

15 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL Proof. Consider one-dimensional auxiliary problem ( Tu 00 + tc()u 0 + u = f (0;x 0 ) u(0) = u(x 0 )=0 or u(0) = u 0 (x 0 )=0 or u 0 (0) = u 0 (x 0 )=0 (5.4) where c() C [0;x 0 ]. When T > t where = kt c(x)k L (0;x0 ), (5.4) then allows a unique solution u H (0;x 0 )by Garding's inequality (cf. Gilbarg & Trudinger[3]) R x 0 0 [T ju0 j + tc(x)u 0 u]dx T R x0 0 ju 0 j dx T t R x 0 0 juj dx. Moreover kuk H (0;x 0 ) Ckfk L (0;x 0 ) where C is independent of the parameter t. Using the above result, we shall construct f^g j g m j= H 3= (); f^h j g m j= H = () from the given g such that f^g j g m j= ; f^h j g m j= satisfy (3.)-(3.5) where ^h j = c()^g 0 j ; 8j S and where ^g j; j S is a solution of (5.4) with respect to f = gj ej, x 0 = je j j, and c() to be determined later with corresponding boundary conditions due to compatibility conditions (.) and (.). Then there is a G V such that G = ^g j ; G = ^h on e j ;j = ; :::; m by theorem.4.6, Grisvard[4] and P the denition of V and kgk V C()[ k^h j k H = (e j ))+ (k^g j k H (e j )+k^h j k H = (e j ))]. js P jd[n (k^g j k H 3= (e j ) + Without loss of generality, we just need to consider (3.)-(3.5) on e j = (A j ;A j ) for j S xed. There are four possible combinations for the neighborhood sides of e j. But no matter what combination it is, we can always take c() as a linear function depended on only the geometry of domain as follows. Case : (j ;j+)dd. Let c(0) = cot! j or 0 if! j = and c(je j j) = cot! j or 0 if! j =. Let ^g j be a solution of (5.) with ^g j (A j ) = ^g j (A j )=0. We let ^h j = c()^g 0 j ()one j. Then we take h j ; h j+ as arbitrary smooth functions such that h j (A j )= csc! j ^g 0 j (A j ) or 0 if! j = and h j+ (A j ) = csc! j ^g 0 j (A j) or 0 if! j =.

16 6 WALTER LITTMAN BO LIU Case : (j ;j +) DN. Let c(0) = cot! j or 0 if! j = and c(je j j) = tan! j or 0 if! j = ; 3. Let ^g j be a solution of (5.) with ^g j satisfying compatibility conditions (.)-(.) at points A j and A j. We take ^h j as in ) and ^g j+ as arbitrary smooth function such that ^g j+ (A j )=^g j (A j ) and ^g 0 j+ (A j)= sec! j^g 0 (A j j) or 0 if! j = ; 3. Similarly one can construct ^g j, ^g j, ^h j and ^h j+ when (j ;j+)nd. Case 3: (j ;j+)nn. Let c(0) = tan! j or 0 if! j = ; 3 and c(je j j) = tan! j or 0 if! j = ; 3. Let ^g j be a solution of (5.) with ^g 0 j satisfying compatibility conditions (.)-(.) at points A j and A j. We take ^g j+ as in ) and ^g j as arbitrary smooth function such that ^g j (A j ) = ^g j (A j ) and ^g 0 j (A j )= cos! j ^g 0 j(a j ). Q.E.D. Remark 5.: theorem 5. still holds when contains more than two consecutive sides and! j =, 8(j; j +)SS. Theorem 5.3. Let > 0 be a constant. If the variational solutions of equations (.a)-(.d) with compatibility conditions (.)-(.) belong to V for any (f; g) L () L ( ), then the variational solution of equations (5.) with compatibility conditions (.)-(.) belongs to V. Proof. From the assumptions made and the method of continuity we know that L t maps V onto L () L ( ) for each xed small t>0. By theorem 5. we know there is G V such that theorem 5. holds. Let u(t) denote the solution of equation L t u = (f; g). Replacing u(t) by u(t) G, we reduce the boundary part t u T u +u = g to the case g =0. By theorem 4. we haveku(t)k V kgk V +ku(t) Gk V C()[kgk L ( ) + kf Gk L ()] C()[kgk L ( ) + kfk L ()] where C() is independent of the parameter t. Letting t! 0. we conclude that the theorem 5.3 holds. Q.E.D.

17 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL Finally we study the singularities of the variational solutions of (.4) for other general polygonal domain. It is easy to check that L t (V ) is a closed subspace of L () L ( ) for any t [0; ]. Let t denote the orthogonal subspace of L t (V ) for every xed t [0; ] in L () L ( ). Lemma 5.. dim( )=dim( 0 ) <. Proof. From the proof of theorem 5. and 5., one has that t ; t [0; ] are the same and therefore have the same dimension and that dim( 0 ) < follows from the results of theorem.3.7 of Grisvard[4]. Q.E.D. If the geometry of the polygonal domain does not satisfy the conditions in corollary 5., then dim( ) > 0. Therefore L can not map V onto L () L ( ) but it does map V onto L () L ( ). We can choose a subspace X of V such that L maps X [V onto L () L ( ). Theorem 5.4. There exist an integer () and linearly independent functions u ; :::; u such that for any given (f; g) L () L ( ), there are unique constant c ; :::; c such that u X i= c i u i V (5.5) where u is a variational solution of (.4) with respect to (f; g). Proof. Let = dim( 0 ) < by lemma 5.. Letting (f ;g ); :::; (f ;g ) be a basis of 0, one takes u ; :::; u as variational solutions of (.4) with respect to (f ;g ); :::; (f ;g ) individually. Then u ; :::; u are linearly independent in V. Thus theorem 5.3 follows from the projection theorem of Hilbert space. Q.E.D.

18 8 WALTER LITTMAN BO LIU We now try to nd an explicit form of such linearly independent functions u ; :::; u such that (5.5) holds. Let r j be the distance from (x; y) in the neighborhood of A j to A j. j (r j ) D(), j = ; :::; m are smooth truncation functions such that j in the very small neighborhood of A j with supp( j ) \ supp( i ) = whenever i 6= j. consider u j;k = j (r j )r j;k j ' j;k () where ' j;k () and j;k are given as follows ' j;k () = ' j;k () = ' j;k () = s s s With running counterclockwise, sin( j;k ); j;k = k if (j; j +)(D[S)(D[S);! j! j sin( j;k ); j;k = (k ) if (j; j +)N(D[S);! j! j sin[(! j ) j;k ]; j;k = (k ) if (j; j +)(D[S)N;! j! j 8 >< >: ' j;k () = ' j; () = s s cos( j;k ); j;k =! j (k )! j ; k ;! j ; j; =0 if (j; j +)NN: Direct calculation shows that u j;k where j = ; :::; m and k is a positive integer belong V and are the variational solutions of (.a)-(.d) with compatibility conditions (.)-(.) with respect to certain f = f j;k D() and g = g j;k L ( ). One also has that (f j;k ;g j;k ) are linearly independent and each (f j;k ;g j;k ) are not orthogonal to 0 when j;k < (cf. x.4, Grisvard[4]). Using the similar argument as in the proof of theorem.4.3 of Grisvard[4] and theorem 5.4, we have Corollary 5.. If 0 <! j <for j =; :::; m, then there exist unique numbers

19 THE REGULARITY AND SINGULARITY OF SOLUTIONS OF CERTAIN ELLIPTIC PROBLEMS ON POL c j;k such that u X ( X j 0< j;k < c j;k u j;k ) V (5.6) for every variational solution u of (.4) with respect to (f; g) L () L ( ). Remark 5.: Under the assumptions of corollary 5., it follows from (5.6) that we have that the variational solutions of (.4) belong to H () for every < + inff j;k ;0< j;k < g. Remark 5.3: It seems likely that the same technique works for more general elliptic equation of second order in and the curved polygons. address of second author: liu math.umn.edu References. [] Adams R., Sobolev Spaces, Academic Press, New York (975). [] Banasiak J. and Roach G. F., On mixed boundary value problems of Dirichlet oblique-derivative type in plane domain with piecewise dierentiable boundary, J. Di. Equ. 79 (989), -3. [3] Gilbarg D. and Trudinger N. S., Elliptic Partial Dierential Equations of Second Order, Springer-Verlag, New York/Berlin (983). [4] Grisvard P., Singularities in Boundary Value Problem, Masson, Paris; Springer- Verlag, Berlin (99). [5] Henry D., Geometric theory of semilinear parabolic equations, Springer-Verlag, New York/Berlin (98). [6] John F., Partial Dierential Equations. 4 th edition, Springer-Verlag, New York (98).

20 0 WALTER LITTMAN BO LIU [7] Littman W., Boundary control theory for beams and plates, Proc. 4 th Conf. on Decision and Control, Fort Lauderdale (985), [8] Littman W., Aspects of boundary control theory, Math. in Sciences and Engineering, Vol 86. \Di. Eq. and Math. Phy."edited by C. Bennewitz, Academic Press Inc., New York (99), 0-6. [9] Mghazli., Regularity of an elliptic problem with mixed Dirichlet-Robin boundary conditions in a polygonal domain, Calcolo 9, No. 3-4 (993), 4-67.